H-LLL: Using Householder Inside LLL

Ivan Morel, Damien Stehlé and Gilles Villard

Abstract: We describe a new LLL-type algorithm, H-LLL, that relies on Householder transformations to approximate the underlying Gram-Schmidt orthogonalizations. The latter computations are performed with floating-point arithmetic. We prove that a precision essentially equal to the dimension suffices to ensure that the output basis is reduced. H-LLL resembles the L2 algorithm of Nguyen and Stehlé that relies on a floating-point Cholesky algorithm. However, replacing Cholesky's algorithm by Householder's is not benign, as their numerical behaviors differ significantly. Broadly speaking, our correctness proof is more involved, whereas our complexity analysis is more direct. Thanks to the new orthogonalization strategy, H-LLL is the first LLL-type algorithm that admits a natural vectorial description, which leads to a complexity upper bound that is proportional to the progress performed on the basis (for fixed dimensions).

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